Elsevier

Computational Materials Science

Volume 109, November 2015, Pages 388-398
Computational Materials Science

New finite element developments for the full field modeling of microstructural evolutions using the level-set method

https://doi.org/10.1016/j.commatsci.2015.07.042Get rights and content

Abstract

Recently a new numerical model devoted to the full field modeling of microstructural evolutions at the polycrystal scale has been proposed and validated [1]. The latter is based on a level set description of interfaces in a finite element framework. Firstly introduced to model 2D and 3D primary recrystallization with nucleaction [1], [2], it has then been extended to consider the grain growth stage [3], [4]. The ability of this approach to model the Zener pinning phenomenon without any assumption concerning the shape of second phase particles was also demonstrated [5]. This model has nevertheless an elevated computational cost and requires many numerical parameters whose calibration is not straightforward. In the present paper, some major improvements of the model which address these two points are discussed. A comparative study is also provided in order to illustrate the gains achieved in terms of computational efficiency and robustness.

Introduction

The mechanical and thermal properties of metallic materials are strongly related to their microstructure. The understanding and the modeling of the microstructural evolution mechanisms is then crucial when it comes to optimize the forming process and the final in-use properties of the materials. Macroscopic and homogenized models, also called mean-field models are widely used in the industry, mainly due to their low computational cost. They are generally based on empirical laws and thus require many fitting parameters which must be calibrated through experimental testing or lower-scale simulations. Furthermore, given the complexity of modern metallurgical problems, these models may not be accurate enough to capture the local but significant events. Abnormal grain growth is an example.

Thanks to the explosion of computer capacities, finer modeling techniques are now available. These lower scale approaches, the so-called full field models, are based on a full description of the microstructure topology [6], [7]. They have demonstrated an interesting potential for the modeling of non-averageable phenomena, such as abnormal grain growth, which cannot be predicted with homogeneized approaches. Over the last decades, several mesoscale numerical models have been developed to simulate the microstructure evolution due to recrystallization (ReX) [8]. Probabilistic voxel-based approaches such as Monte Carlo [9], [10] (MC) and cellular automata [11] (CA) are very popular. There are also deterministic approaches, which are more precise, as they do not rely on probabilistic laws, but also, of course, more greedy in terms of computational resources due to the fact they involve the solving of large systems of partial differential equations (PDEs). Thus several workers have developed the vertex method [12] wherein the grain boundaries are defined in terms of vertices; the interface motion is then imposed by the displacement of a set of points. A major limitation of these front-tracking approaches lies in the handling of topological events (grain shrinkage, appearance of a nucleus), especially in three dimensions. Another approach found in the literature is the phase-field (PF) method, which offers the advantage of avoiding the difficult problem of tracking interfaces [13]. In these models, the interfaces evolve to minimize a thermodynamic potential of the system. Many different forms can be found in the literature for this potential [14], [15], [16], [17] because any function satisfying certain mathematical properties can be chosen. A comprehensive review can be found in [18]. Finally, grain growth (GG) and ReX can also be modeled using a level set (LS) description of the interfaces in the context of uniform grids with a finite-difference formulation [19], [20] or in a finite element (FE) framework [3], [4], [5], [21]. The FE-LS and FE-PF methods are actually quite close and share some common features (no explicit tracking of the interfaces, deterministic approach lying on the solving of PDE systems). Although the PF method lies on strong physical and thermodynamical foundations, its formulation introduces purely numerical parameters (like the grain boundary width). On the other hand, the LS method only requires measurable quantities which have a direct physical interpretation, making it more simple to use. The comparison of these two methods is a quite new but motivating topic in the literature. Recently, a comparative discussion concerning the modeling of GG with anisotropic boundary energy was proposed in [22]. This topic is also discussed in [20]. It must be highlighted that the recoloring scheme proposed in the third section of the present paper could also be employed in a FE-PF context. Interest of LS method in a FE framework lies also on the possibility to handle large polycrystal deformations with the well-known crystal plasticity finite element method (CPFEM) [2], [23].

The full field modeling of ReX using unstructured FE meshes is a very exciting research topic due to the possibility to model and predict many physical phenomena (particle pinning, annealing twin development, solute drag, CPFEM and field dislocation mechanics, …). However, the numerical cost of these approaches remains their main drawback, explaining why they are hardly used for 3D computations.

The present work addresses this issue. After introducing the FE-LS numerical model in the next section, some recent improvements are presented in the third chapter. A comparison of the old and current configurations and results of different 2D and 3D large-scale simulations are finally provided in the fourth section. More specifically, a 3D LS simulation of the particle pinning phenomenon without any assumption concerning the interactions between the second phase particles (SPP) and the grain boundary is described.

Section snippets

Numerical model, applications and related issues

As mentioned above, the model considered in this paper works around a LS description of the interfaces in a FE framework. A LS function ψ is defined over a domain Ω as the signed distance function to the interface Γ of a sub-domain G of Ω. The values of ψ are calculated at each interpolation point (node in the considered P1 formulation) and the sign convention states ψ>0 inside G and ψ<0 outside:tψ(x,t)=±d(x,Γ(t)),xΩ,Γ(t)={xΩ,ψ(x,t)=0},where d(.,.) corresponds to the Euclidean distance.

Development of an efficient grain recoloring algorithm

As mentioned above, the objective is here to handle dynamically the distribution of the grains inside the GLS functions. The algorithm must then be able to transfer automatically one or more grains in another GLS functions if child grains are becoming too close from each other. The advantage of such an approach lies in the fact that it enables to limit the number of needed functions while avoiding coalescence in the same time. Furthermore, it allows to set the parameter δ to a constant value

Comparative study and large-scale simulations

In this chapter we are interested in demonstrating the efficiency of the newly developed tools and the gains achieved in terms of numerical efficiency and robustness. A comparative study between the old and current situations is provided and different 2D/3D large-scale simulations which would not have been performed in acceptable computation times without these improvements are detailed. In the following, one mainly discusses the numerical aspect of these simulations (computational efficiency

Conclusion

A new full field model devoted to the modeling of microstructural evolutions based on the LS method in a FE framework is discussed. This model has already been used to simulate primary recrystallization and GG with possible inert second phase particles and shows good agreement with experimental observations and other models. Nevertheless it suffers from high computational costs and requires many numerical parameters whose calibration is not straightforward. In this paper, an algorithm enabling

Acknowledgements

The authors would like to gratefully acknowledge Transvalor for funding this reseach.

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